Statistical Mechanics in the Label Framework: Entropy as a Label Count, the Second Law as Initial Data

In the label framework, matter is a field of comoving labels; a relabelling with unit Jacobian leaves the label density and every term of the action unchanged. We ask what labels give in statistical mechanics. For many discrete labels, entropy relative to the label volume is, per label, the logarithm of a count of arrangements. Flows that preserve phase-space volume, as the Vlasov flow does, keep every functional ∫f of the density and so the fine-grained entropy. In lattice checks coarse-grained entropy grows with coarse-graining, mixing and low-entropy initial data; the inverse flow undoes it. The second law follows neither from the symmetry nor from volume-preserving dynamics alone; we read its arrow as initial data. Diffusion is the Wasserstein gradient flow of the negative entropy; its entropy production is D times the Fisher information of the quantum term. Maximum entropy gives the Gibbs form, with temperature a label.